Question

There is a horizontal cylindrical uniform but time varying magnetic field increasing at a constant rate $\frac{dB}{dt}$ as shown. A charged particle having charge q and mass m is kept in equilibrium, at the top of a spring of spring constant K in such a way that it is on the horizontal line passing through the center of the magnetic field as shown in figure. The compression in the spring will be

• A. $\frac{1}{K}[mg-\frac{q{R}^2}{2l}\frac{dB}{dt}]$
• B. $\frac{1}{K}[mg+\frac{q{R}^2}{l}\frac{dB}{dt}]$
• C. $\frac{1}{K}[mg+\frac{2q{R}^2}{l}\frac{dB}{dt}]$
• D. $\frac{1}{K}[mg+\frac{q{R}^2}{2l}\frac{dB}{dt}]$

Answer 1

The correct option is D $\frac{1}{K}[mg+\frac{q{R}^2}{2l}\frac{dB}{dt}]$

$E2\pi l=\pi .R^2\left(\frac{dB}{dt}\right)$
$E=\frac{R^2}{2l}\left(\frac{dB}{dt}\right)$
$qE+mg=Kx$
$x=\frac{q.R^2}{k2l}\left(\frac{dB}{dt}\right)+\frac{mg}{K}$
$x=\frac{1}{K}[mg+\frac{q{R}^2}{2l}\frac{dB}{dt}]$